Given a geodesic triangle and let . We ask how far from the other sides is ? Well, inscribe a semi-circle centered at inside of ; pick the largest such inscribed semi-circle, and call its radius . So is -slim, where is the largest ; that is, is the radius of the largest semi-circle that can be inscribed in .

So to find , we look at semi-circles; for this, we need a fact about .

Fact. For any , , where are angles of the triangle.

This leads to a uniform bound on the area, and hence the radius of semi-circles inscribed in .

To define hyperbolic groups, we want to prove hyperbolicity is a quasi-isometric invariant of geodesic metric spaces. We need to “quasi-fy” the definition of -hyperbolic.

Definition. A quasi-geodesic is a quasi-isometric embedding of a closed interval.

Exercise 13. Let by in polar coordinates. Show that is a quasi-isometric embedding.

We will prove this behavior does not happen in hyperbolic metric spaces.